L’INSTITUTFOURIER DE ANNALES

L’INSTITUT FOURIER

David HARARI & Alexei N. SKOROBOGATOV

The Brauer group of torsors and its arithmetic applications Tome 53, n^o7 (2003), p. 1987-2019.

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THE BRAUER GROUP OF TORSORS AND ITS ARITHMETIC APPLICATIONS

by

^{D. HARARI and A. N.} SKOROBOGATOV

Introduction.

This _paper consists of two

parts.

^{In the first}

part

^{we describe the} behaviour of the Brauer _group with

respect

^{to the}

pull-back

map from the

variety

^{to a torsor under a torus over} any field of characteristic 0.

To a

variety

^X endowed with a

surjective morphism

^{: X ---+}

P~

^with

geometrically integral generic

^{fibre one}

canonically

associates certain X-

torsors,

called vertical torsors. The second

part

^{of the} paper ^concerns the arithmetic of vertical torsors over a number field k. Let K ⁼

k(t)

^{be the}

function field of

P’,

and let K be an

algebraic

closure of K. Assume that the Picard _group of the

geometric generic

^fibre

XK

^is

finitely generated,

^and

that its Brauer _group is finite. These conditions are

satisfied,

^for

example,

when

XK

^is

rationally

connected. We _prove that the Manin obstruction to the Hasse

principle

^{and weak}

approximation

^on smooth and _proper models of such torsors is the

only

^one

provided

^{the same}

property

holds for the k-fibres of _7r, and the number of ’bad’ fibres is small.

Using

the results of the first

part

^we

give

sufficient conditions for X itself

(strictly speaking,

^for

any smooth and _proper model of

X)

^{to have the} afore-mentioned

property.

To illustrate

possible applications

^{we exhibit}

apparently

^new classes of conic bundle threefolds and some other varieties with the

property

^{that the}

Keywords: ^Brauer group - Hasse principle - ^{Universal torsor.}

Math. classification: llG35 - 14G05.

f :

^{Y - X}

variety

^X such that all invertible functions on X are

constants,

^{and Pic X has no}

torsion, they

prove that the kernel of the natural _map Br Y - Br Y is

naturally isomorphic

^{to Br k. In this} paper

we show

(Theorem 1.7)

^{that the} map Br X -

Br Y/Br k

^defined

by f

^is

canonically

identified with the _map Br X -

(Br X )r (provided

^{k is such}

that

H3 (r,1~* )

⁼

0).

We also obtain sufficient conditions that ensure that the unramified Brauer _group of Y is the

image

of the unramified Brauer group of X.

In the arithmetic

part

^{of the} paper ^we

present

^a

generalization

^{of the}

results of

[H94],

^{as well as of those of}

[S90], [S96], [CSOO],

that allows a

few bad fibres in the theorem on the behaviour of the Manin obstruction

regarding

^{the passage from} the k-fibres to the total _space. Our main achievement are Theorems 2.12 and

2.9, proved by

^a combination of

’open

descent’ and fibration methods. Let X be a smooth and

projective variety

over a number field

k,

^{and let X -}

P~

^{be a dominant}

^morphism

^with

geometrically integral generic

fibre. Assume that the sum of

degrees

^{of the}

closed

points

^of

Pk corresponding

^to

non-split (bad)

^{fibres is at most}

^3,

^and

that _every

non-split

^{fibre contains a}

multiplicity

¹ irreducible

component splitting

^{into two over an}

algebraic

closure of the residue field. Assume further that

PicXK

^is

finitely generated, Br XK

^is

^finite,

^and

^XK

^{has a}

k(t)-point.

^{Then we} prove that unless we are in an

explicitly

^described

exceptional

case, the

possible

failure of the Hasse

principle

^{and weak}

approximation

^{on X is} accounted for

by

the Manin obstruction whenever the same is true for smooth k-fibres of 1T. We also _prove a similar result in the case when there are

only

^two

non-split fibres,

^{both over}

k-points,

each

containing

^an irreducible

component

^of

multiplicity

^{1 defined over an}

extension of k of

prime degree. Explicit examples

^of

applications

^{can be}

found in the end of the _paper.

Notation and conventions

All

cohomology

groups in this _paper are either _group

cohomology

^of

finite _groups, Galois or 6tale

cohomology

groups.

(For example,

^{we write}

7~(~,...)

^for

H6t(X,...)

^{for a scheme}

X.)

^{Let F be a finite} group, and M be an F-module. Then

M)

denotes the

subgroup

^of

M)

^con-

sisting

of the classes whose

image

under the restriction map to every

cyclic subgroup

^{of F is zero. For a discrete} r-module M we write

M)

^{for the}

Galois

cohomology

group

Hi (r, M).

Assume that

M,

^{as an abelian} group, is of finite

type

and torsion-free. We define for i ⁼

1, 2,

^{where G is the}

(finite) ^image

of the action of r on M. Note that the groups

M),

^{i =}

1, 2,

^are finite. Let 1~’ be the smallest extension of k such that the action of Gal

(k/k’)

^{on M is} trivial. Write G ⁼ Gal

(k’lk).

Then

H’(k’, M)

^{= 0 because}

^HI (k’, Z)

^{= 0.} The restriction-inflation se-

quence for M

gives M) = Hl (G, M).

^{There is also an} exact sequence

We conclude that

M)

^is

naturally

^a

subgroup

^of

M).

When 1~ is a number field we denote

by Qk

^{the set of}

places

^of

k,

^and

by kv

^the

completion

^{of k at the}

place

^{v. Let}

Ak

^{be the}

ring

of adeles of k. It is an easy

corollary

of the Tchebotarev

density

theorem that for a

number field k the _group

M)

consists of a E

M)

^{such that}

the localisation cxv E

M)

^is trivial for almost all

places

^{v E}

Ok.

By

^a

variety

^{X over a field k we} understand in this _paper a

separated

scheme of finite

type

^over

Speck.

The Brauer _group Br X ⁼

H2 (X, Gm)

is

equipped

^{with a} natural filtration

Bro X

^C

Br1 X

^C

Br X,

^where

BrX]

and Set

A;[X]*

⁼

HO(X, Cm ) .

^{There is a} Hochschild-Serre

spectral

sequence

J

It

yields

^{a canonical} map When we

obtain the exact sequence of low

degree

^terms

and a

complex

Recall that for _any number field k the local invariant of class field

theory

^{is an}

injective map jv : Br kv

^{-~ which is an}

isomorphism

^for

finite v. Let X be a proper, smooth and

geometrically integral k-variety.

Then the set of adelic

points X(Ak)

^{is the}

product TIvEOk

^Assume

that

0,

^{and let}

X(k)

^{be the} closure of the

diagonal image

^of

X(k)

in

X(Ak ) .

^Set

(Note

^{that the sum is well}

^defined,

^see

[CS87a]

^{III or}

[SO1], 5.2).

^The

reciprocity

^{law of}

global

^{class field}

theory implies

^that

In

particular,

the condition

0

^{is an} obstruction to the existence of a k-rational

point

^on

X;

this is the Manin

(or Brauer-Manin)

obstruction to the Hasse

principle.

The condition

X (Ak)

is the Brauer- Manin obstruction to weak

approximation.

^One says that the Brauer- Manin obstruction to weak

approximation

^{is the}

only

^{one if}

X(k) =

The

property

that the Brauer-Manin obstruction to the Hasse

principle (resp.

^{to weak}

approximation)

^{is the}

only

^{one is a} k-birational invariant of smooth and _proper varieties.

(This

^follows from the birational invariance of the Brauer _group

([G], 6.2),

^{the v-adic}

implicit

^function

theorem,

and Nishimura’s

lemma,

^which says that the condition

X (1~) ~ ~

is a k-birational invariant of smooth and _proper

varieties.)

1. Torsors

and the

Brauer group.

In this section k is _any field of characteristic zero.

1.1. Torsors under tori and their relative

compactifications.

By

^{T we} shall denote an

algebraic

k-torus with module of characters

T.

Let _{p :} X 2013~

Speck

^{be a} smooth and

geometrically integral k-variety,

and let

f :

V 2013~ X be a torsor under T.

Equivalently,

^{Y is a smooth and}

geometrically integral k-variety equipped

^{with a}

(scheme-theoretically)

^free

action of T such that

YIT

^{= X.}

We shall

frequently

^{use the}

Leray spectral

sequence

By

the relative Rosenlicht lemma

([CS87a], Prop. 1.4.2)

^{we have an exact}

sequence of sheaves on X in the 6tale

topology:

Here

p*T

^{is a}

sheaf

on X obtained from the F-module

T.

Note that

Z)

^{= 0 for any normal}

integral

^{scheme S}

([SGA 4],

^{IX 3.6}

(ii)j.

This

implies

ⁱⁿ

particular

^{that = 0.}

LEMMA 1.1. ^- Let X be a normal

k-variety,

^{and let}

f :

^{Y ---+ X be a}

torsor under ^a torus T. Then we have the

following properties:

Proof.

(i)

is established in the

proof

^of

([CS87a], Prop. 2.1.1 ) .

^For

the convenience of the reader ^we

reproduce

^this

argument

^{here in the case} T ⁼

G, alongside

^{with the}

proof

^of

(ii).

Let x be a

geometric point

^of

X,

^{R be the}

strictly

henselian local

ring

of X _{at x,} and K be the field of fractions of R. The stalk of at

x is

Pic (Y

^{x x}

R),

and the stalk of

R2 f*Gm

^{at x is Br}

(Y

^Xx

R).

^Note

that

Gm)

^{= 0 for} any local

ring

^R

([M], III.4.10),

hence the scheme Y _{x x} R is

isomorphic

^to

Gm, R.

We have Pic

Gm,s -

^{Pic ,5’ for any normal}

base scheme S. Now 0 which

implies Pic (Y

^{x x}

R) =

^0.

Therefore, RI f*Gm

^{= 0.}

Recall that a

ring

^{of finite}

type

^{over an excellent}

ring (resp.

^the

localization of an excellent

ring)

^{is excellent}

( ~EGA4~ , 7.8.3).

Hence the local

ring Ox,x

^{is an excellent}

^ring.

^{Since k is of} characteristic _zero, Hironaka’s resolution of

singularities

^{holds for}

Ox,x. By [EGA4],

^{7.9.5 and}

^18.8.17,

^the

From

(2), (3)

^{and statement}

(i)

of this lemma we obtain an exact sequence off-modules

([CS87a], Prop. 2.1.1,

^{see also}

[SO1], 1.3):

The _map

T -

Pic X in this exact sequence is called the

type

^{of the torsor}

f :

^{Y - X. The torsor}

f :

^{Y - X is} called universal if its

type

^{is an}

isomorphism.

^{In this case we} have Pic Y = 0. When ⁼

1~*

the

type

of the torsor

f :

^{Y - X is}

injective

^{if and}

only

^if

~[V]*

⁼

^k*.

^In

particular,

this

property

holds for universal

torsors,

^and

implies Bri

^{Y =}

Bro

^{Y = Br k}

by (1).

In this _paper we define a relative smooth

compactification

^{of a torsor}

under a torus as follows.

DEFINITION 1.2. - Let

f :

^{Y -~ X be a torsor under a torus. A}

relative smooth

compactification

^of

f :

^{Y - X is a}

geometrically integral variety

^Z endowed with a smooth and _proper

morphism g :

^{Z - X with}

connected

geometric fibres, together

^{with an}

embedding

^{i : Y - Z such}

that

i(Y)

^{is open and dense in}

^Z,

^and

f =

g ^o i.

The existence of a relative smooth

compactification

ⁱⁿ characteristic 0 follows from Hironaka’s theorem.

DEFINITION 1.3

[CS77].

^{- An} exact sequence off-modules

such that

M, P, F

^{are free abelian} groups of finite

rank,

^{is called a}

flasque

resolution of M if the

following

conditions hold:

(a)

^{P is a}

permutation r -module,

^that

is,

^{it has a} r-invariant Z-basis.

(b)

^{F is}

flasque,

^that

is,

^{the Tate}

cohomology

group

F)

^is

trivial for _any

subgroup

^{G’ C}

G,

^{where G is the}

image

of the action of r

on F.

LEMMA 1.4. ^- Let X be ^a smooth

k-variety,

^{and let}

f :

^{Y --~ X be a}

torsor under a torus T of

type

^{T :}

T -

Pic X.

Let g :

^{Z - X be a relative}

smooth

compactification

^of

f :

Y - X. Then

(i)

^{there is a natural}

^flasque

resolution

of T :

Proof - We _prove

(i)

^and

(ii) simultaneously.

^{Since Z is}

regular

^we

have an exact sequence of sheaves in the 6tale

topology

^{on Z:}

The sheaf

DivzBy

^{is defined as} the direct sum of

j,,Z, where j

^{is the}

embedding

^{into Z of a closed}

integral

subset of codimension 1 contained in

Z B Y,

for all such closed subsets of Z. The _sequence

(5)

^is exact since Z is

regular,

and hence the Weil divisors coincide with Cartier divisors

(see [M], ^{II, Example} 3.9).

^Let

D+ (X) (resp. ’D+ (Z))

be the derived

category

^of bounded below

complexes

of 6tale sheaves on X

(resp.

^on

Z).

The derived functor

I~g* : D+ (Z)

^-~

D+ (X ) ^gives

^{an exact}

triangle

ⁱⁿ

D+ (X) :

In

D+ (X)

^{there is also an exact}

triangle

^defined

by (3).

^{There is a natural}

isomorphism

⁼ The canonical

morphism

^-1

is an

isomorphism

^{in our case}

since g

^is proper with connected fibres. These two

morphisms

^{define a}

morphism

^{of exact}

triangles

ⁱⁿ

D+(X):

Let K be the function field

k(X), ^{Ko -} k(X).

We have the

following

identifications:

From

(6)

^we obtain the

following

commutative

diagram

of r-modules with exact rows and

columns,

^{which is the} definition of F:

The arrows in the

right

^hand square ^are the natural _maps. The

top

^row

agrees with the exact sequence

(4);

^{the map} bi~ - Pic X is the

type

^{T of the}

torsor

f :

^{Y - X. We} observed that

Gm,x - g*Gm,z.

^This

^implies

^that

k[X]*

⁼

k[Z]*,

and that Pic X

injects

into Pic Z

(by

^the

Leray spectral sequence).

Now it follows from the

diagram

^{that the} map

T

^-~

is

injective.

^This proves the exactness of the sequence in

(i).

To

complete

^the

proof

^of

(i)

^{we now} show that F is a

flasque

^r-

module. Let k’c k be the smallest

(necessarily Galois)

^{extension of k such}

that Gal

(k/k’)

^acts

trivially

^on the modules in

diagram (7).

^{Call G the}

finite _group Gal

(k’lk).

Note that the

generic

^fibre

ZKo

^smooth

compactification

^{of the}

principal homogeneous

space

YKo

^{= Y x x}

^Ko

^{of the}

torus T _{x x}

Ko.

^We

identify

^{r = Gal with Gal}

(K/Ko ) .

^Then

we have a canonical

isomorphism

^{of r-modules}

T (see,

^e.g.

[SOl],

^Lemma

2.4.3).

^{This leads to an} exact sequence of r-modules

The exact sequence

(8)

^{will not}

change

^{if we}

replace

the extension K ⁼

k(X)

^of

Ko

⁼

k(X) by

the extension L ⁼

k(Y) ^{of Lo}

⁼

k(Y). ^Indeed,

^since

Ko

^is

algebraically

^{closed in}

Lo

the Galois _group Gal

(L/Lo )

^is

canonically isomorphic

^{to r = Gal}

(K/Ko ) .

On the other

hand,

^{we are}

dealing

^{with r-}

modules which are free abelian _groups of finite

type

^{that do not}

change

^after

the _passage from K to L. The

advantage

^{now is that Y has a canonical}

Lo- point given by

^the

embedding

^{of the}

generic point

^into Y. Hence the Galois module Pic

ZL

^{= Pic}

ZK

^is the Picard module of a smooth

compactification

of the torus

TL.

^{The Galois} group r acts on

Pic ZL

^{via its}

quotient

^G.

By

a theorem of Voskresenskii

([V],

^Thm.

4.6)

^{the G-module}

^{Pic ZL}

^is

^flasque.

The G-module Div is the direct sum of the module of ’vertical’

divisors

Div v,

^that

is,

the divisors on Z that do not intersect the

generic

fibre

ZK,

and the module of ’horizontal’

divisors,

^that

is,

the divisors ^on Z that are Zariski closures of the divisors on

Zx

^with

support

ⁱⁿ

Zx B

The G-module of horizontal divisors is thus identified with Div ZK B YK Restriction to the

generic

fibre defines a map of G-modules Pic Z - Pic This _{map is}

surjective

^{because Z is}

regular.

Its kernel

obviously

^contains

Pic X,

^{hence it factors}

through

^a

surjective

map F --~

Pic Z K .

The

composition

^{of the} map T -~ from

diagram (7)

^with

the _map ^--~ Div

(Zx) given by

the restriction of divisors to the

generic

^fibre

Zx,

^{is none} other than the second _map from

(8). ^Indeed,

the

right

hand vertical arrow in

(6) gives

^{rise to a} commutative

diagram

which is

precisely

^{what we need:}

Putting everything together

^{we obtain an exact} commutative

diagram

^of

G-modules

(The

^exactness of the bottom row follows

immediately

^{from the exactness}

of the columns and the

top row).

^Since

Div v

^{is a}

permutation module,

^{it is}

flasque by Shapiro’s

lemma. Now the bottom line of the

diagram

^{shows that}

F is

flasque (in fact,

^an extension of a

flasque

^module

by

^a

permutation

module is

split).

This establishes

(i).

In

particular,

^{we have}

~I - ~ (C, F) -

^{0 for} any

cyclic subgroup

C C G.

By

^the

periodicity

^{of Tate}

cohomology

^of

cyclic

groups ^we obtain

H 1 (C, F)

^{= 0 for} any

cyclic subgroup

^{C C G.}

We now go back to

diagram (7).

^The

just

established

property

^of

F,

and the fact that

DivZBY(Z)

^{is a}

permutation G-module,

^{and hence}

enable us to conclude

that

coincides with the

injective image

of in

H2(1~, T) (cf. [CS87b],

^the

proof

^of

Prop.

^9.5

(ii)).

^The

statement

(ii)

^now follows from

diagram (7).

^D

Remark 1.5. - We have associated a canonical

flasque

^resolution

of

T

_{to any} relative smooth

compactification

^{of an} X-torsor under T.

This

generalizes

the construction due to Voskresenskii and Colliot-Thelene- Sansuc

(see [V], [CS77])

when the X is the

spectrum

^{of a field.}

1.2. The

pull-back

of the Brauer _{group to} a torsor.

In connection with arithmetic

applications

^{of torsors} under tori we are interested in

describing

^the cokernel of the _map

f * :

^{Br X ---+}

Br Y,

especially

^{when Y is a universal torsor.}

Arguably

^{a more essential}

question

is the

computation

of the cokernel of the _map

f nr : Brnr

^-

Brnr

Colliot-Thélène and Sansuc

pointed

^out that when 1~ is

algebraically

^{closed the} map

f nr

^{is an}

isomorphism. (This

follows from the fact that in this situation Y is

birationally equivalent

^{to the}

product

Our first result concerns the cokernel of

f * :

Br X - Br Y over an

algebraically

closed field.

THEOREM 1.6. ^- Let k be an

algebraically

closed field of charac- teristic 0. Let X be a smooth and

geometrically integral variety

^{over k}

such that ^- k* and Pic X is a free abelian _group of finite

type.

^Let

f :

^{Y --~ X be a torsor under a} k-torus such that

k[Y]*

⁼ k* and Pic Y is

a free abelian _group of finite

type (for ^example,

^{a universal}

torsor).

^Then

the _map

f * :

Br X - Br Y is an

isomorphism.

Note that after

[CS87a]

^the

injectivity

^of

f * :

Br X - Br Y is known to be true

(over

^an

algebraically

^closed

field)

^for any torsor under a

torus,

but the

surjectivity

^{does not hold in}

general.

A well-known

example

^{is the}

torsor under but Br

Proof of Theorem 1.6. - Let X be a smooth and

geometrically

connected

variety

^{over an}

algebraically

^{closed field}

k,

^{such that}

k[X]*

^{= k*}

and Pic X is torsion-free. Let

f :

Y --4 X be a torsor under T. We denote its

type by

Ty. The

assumptions

of Theorem 1.6 are

equivalent

^to the condition that _TY is

injective

^and

Ty (T)

^{is a}

^primitive

sublattice of Pic X.

Let us choose ^a Z-basis in

T ~ _Z’~,

and factor

f :

^{Y - X into a}

composition

^{of torsors under}

where each

Yi

^{is a}

k-variety.

^{Then we have}

H° (Y2, Gm)

^{= k*. It is also}

clear that

Pic Yi

^{is a} torsion free abelian _group of finite

type.

^{It is therefore}

enough

^to prove Theorem 1.6 when T =

Gm.

^{We assume this from now on.}

The

spectral

sequence

(2)

and Lemma 1.1

imply

that BrY =

H2 (X, f * Gm ) .

^{Now the} exact sequence

(3) together

with the remark that follows

it,

and the fact that Br Y is a torsion _group lead to an exact se-

quence

now follows from

(10)

^{that the} map

f * :

^{Br X - Br Y is an}

isomorphism.

Theorem 1.6 is

proved.

^D

Over

general

^fields the situation is more

complicated.

THEOREM 1.7. - Let k be a field of characteristic 0 such that

H3 (1~, I~* )

⁼ 0. Let X be a smooth and

geometrically integral k-variety

such that

I~~X~* - ^k*

and Pic X is a free abelian _group of finite

type.

(a)

^Let

f :

^{Y - X be a torsor under a k-torus}

T,

^{such that}

I~ ~Y~ *

^k’

and Pic Y is a free abelian _group of finite

type (for example,

^{a universal}

torsor).

^{Let T :}

T

^-~ Pic X be the

type

^of

f :

Y - X. Let

P,

^{be the}

intersection of the

image

^of

(Br X ) r -~ H2 (I~, Pic X )

^with

T*(H2(k, T ) )

ⁱⁿ

Then there is an exact sequence of abelian _groups

(b)

^Let

f :

^{Y - X be a universal torsor.} Then there is ^a natural

isomorphism

⁼ such that the canonical _map Br X -

(Br X ) r

identifies with the _map Br X -

Br Y/Br k

^defined

by f * .

Note

that,

^{in the}

assumptions

^{of this}

theorem,

^the

surjectivity

^{of the}

map

f * :

Br X - Br Y for a universal torsor

f :

^{Y - X is}

equivalent

^to

the

surjectivity

^{of the} map Br X -

Proof. ^- The natural _map of r-modules

f* :

Br X ~ Br Y is

an

isomorphism by

Theorem 1.6. Hence

isomorphism.

is also an

We now

exploit

^the

functoriality

^{of the} Hochschild-Serre

spectral

sequence

with

respect

^{to the}

morphism f :

Y - X. In view of our

assumption

H3 (1~, l~* )

^{= 0 this}

spectral

sequence

gives

^{rise to the}

following

commutative

diagram

^{with exact rows:}

The statement

(a)

^now follows from this

diagram

combined with exact sequence

(4).

^{To prove}

(b)

^we observe that for a universal X-torsor Y one has Pic Y ⁼ 0. From the

diagram

^{we obtain}

Br Y /Br 1 Y == (Br Y)r,

^and

the result follows. Theorem 1.7 is

proved.

^D

For the sake of

completeness

^{we record a}

proposition

^{which can}

be

proved along

^{the same lines} but without ^{recourse to} the condition

H3 (k, k*)

^{== 0. It}

improves

^a similar earlier result of Colliot- Thélène and Sansuc valid for smooth

compactifications

of universal torsors over

projective

^varieties.

PROPOSITION 1.8. - Let k be ^a field of characteristic 0. Let X be a

smooth and

geometrically integral k-variety

^{such that}

1~ ~X ~ * = 1~* ,

^{Pic X is}

a free abelian _group of finite

type,

^and

(Br

= 0. Then for _any universal X-torsor Y we have Br Y ⁼ Br k.

The

following theorem,

^{in which the torsor is}

replaced by

^{a relative}

smooth

compactification,

^{seems to be more useful in}

applications.

THEOREM 1.9. - We

keep

^{the notation} and conditions of Theorem 1.7

(a).

^Let

QT

^{be the} intersection of the

image

^of

(Br X )r

^-

^H2 (k,

^Pic

X )

with T*

(EL12 (k, T))

ⁱⁿ

^H2 (k,

^Pic

X).

^{Then for any} relative smooth

compact-

ification Z - X of Y --4 X there ^{is an exact} sequence of abelian _groups

Proof. The

beginning

^{of the}

proof

^{is similar to the}

proof

^of

Theorem 1.7.

Let g :

^{Z ~ X be} any relative smooth

compactification

of

f :

Y - X. This time we use the

functoriality

of the Hochschild-Serre

spectral

sequence

with

respect

^{to the}

morphism g :

^{Z - X.}

The natural _map

g* : (Br X )r -~ (Br Z)r

^{is an}

isomorphism. Indeed,

by [G]

^the map i* : ^: Br Z - Br Y is

injective (since

^{Y C Z is an open}

The

proof

^{is now}

completed by

^an easy

diagram

^{chase and an}

application

of Lemma 1.4. D

For a proper

variety

^{X we}

obtain,

^{as a}

corollary,

^a result about the cokernel of the _map Br.

COROLLARY 1.10. - In the

assumptions

of Theorem 1.7

(a)

^assume

further that X is _proper. Then ^we have an exact sequence

where

The

corollary

follows from Theorem 1.9 because in this case Z is a

smooth and _proper model of

Y,

hence Br Z ⁼

Brnr by [G],

^6.2.

Since

0,

^{we have}

Brk,

^{whence the}

following corollary:

COROLLARY l.ll. In the

assumptions

of Theorem 1.7

(a)

^as-

sume further that X is _proper and Y is a universal X-torsor. Then

Brnr

⁼

Br X/Bri

^{X whenever Pic}

X) -

^0.

In the case of a number field k we have a more

precise

^result.

PROPOSITION 1.12. - Let X be ^a _proper, smooth and

geometrically

integral variety

^{over a} number field k such that Pic X is a free abelian group of finite

type.

^{Let Y be a} universal X-torsor. Then the

quotient

^of

the unramified Brauer _group

Brnr (k(Y)/k) by

^the

image

^{of Br X is the}

intersection

where and v _ranges ^over

the

places

^v of k such that the r-module Pic X is unramified at v

(the

inertia group at v acts

trivially).

Proof. The

functoriality

of the Hochschild-Serre

spectral

sequence with

respect

^to restriction from k to

kv gives

^{rise to a} commutative

diagram

with

injective

horizontal arrows:

It follows from the Tchebotarev

density

theorem that Pic

X)

^is

the intersection of the kernels of the restriction _maps

H2 ( k, Pic X) ---+

H2(kv,PicXv)

^{for v} such that the r-module Pic X is unramified at v.

Now the

diagram

shows that

Qid

^is

given by

^the

displayed

^{formula of the}

proposition.

On the other

hand,

^{for a} universal X-torsor Y - X we have

0,

whence the

following

exact sequence:

The

proposition

^{follows. D}

It would be

interesting

^to

give

^an

example

^{of a}

variety

^X

satisfying

the

assumptions

^{of this}

proposition

^{such that}

Brnr

^{contains an} element that does ^{not come} from Br X.

Remark 1.13. ^- If

Y(I~) ~ 0,

then the existence of a section of the map

H3 (1~,1~* )

^-

H3 (y, Gm)

^{makes the}

assumption H3(k, k*) ==

^{0 in}

Theorem 1.7

superfluous.

^{The same is true} for Theorem 1.9 if

Z(1~) ~ ^0.

2.

Arithmetic

of

vertical

torsors.

In this section we cross the fibration method of

[H94]

^and

[H97]

^with

the descent method of

~590~, [S96]

^and

[CSOO].

2.1. Main theorem.

Let 1~ be a field of characteristic

0,

^{and X be a}

geometrically integral

k-variety equipped

^{with a}

surjective morphism f

^to

P~ . ^{Let q}

^{be the}

^generic

over the residue field

k(P)

^{of P.}

LEMMA 2.1. Let X be a

smooth, projective

^and

geometrically integral k-variety.

^Let

f :

^{X - be a}

morphism

^with

geometrically integral generic

fibre which contains a Then there exists a dense open subset U C X such that

(a)

the restriction of

f

^{to U is a smooth}

surjective morphism

^with

integral

^closed

fibres ;

(b)

^{if for a closed}

point

^{P E}

pl

^{the fibre}

^Xp

^is

^split,

then the fibre

Up

^is

geometrically integral;

(c)

^the

generic

^fibre

UK

coincides with

XK.

The ^same conclusion holds under a weaker

assumption

^that

f :

^{X -}

P~ locally

^{in etale}

topology

^{has a section} at every closed

point P

^G

P~.

Equivalently,

every closed fibre of

f

^{contains an} irreducible

component

^of

multiplicity

^1.

Proof. For _any closed

point

^{P of}

P~

the fibre

X p

^{contains an} irreducible

component

^of

multiplicity

^1.

Indeed, XK

^{has a}

k(71)-point

^and

is proper, thus

f

^{has a}

k-section.

Since X is

regular,

the intersection

point

of this section with

X p

^{must be}

regular

^on

Xp.

^In

particular,

^this

point belongs

^{to an} irreducible

component

^of

multiplicity

^1.

For _every closed

point

^{P choose an} irreducible

component Ep

^C

Xp

of

multiplicity

^1. Whenever there is a

geometrically

irreducible

component

we choose it as

Ep.

^Let

Fp

be the union of all the

remaining components

of

X p ,

^{and let}

G p

^{be the}

singular

^{locus of}

Ep.

^The

complement

^{U to the}

(finite)

^{union of the}

Fp

^{and the}

Gp

for all the closed

points

^{P E}

P~

^has

the

required properties.

^D

Let

A r-module M such that M ⁼

MU

^{for a dense} open subset U C X

satisfying

the conditions of Lemma 2.1 will be called a vertical module associated to

f.

A U-torsor of

type Mu -

^{Pic U is} called vertical. It is easy ^{to see} that

MU

^{is a} torsion-free abelian _group

([S96], Prop. 3.2.3).

THEOREM 2.2. ^- Let X be a

smooth, projective

^and

geometrically integral variety

^{over a} number field

k,

^{and let}

f : X - P~

^{be a}

dominant

morphism

^with

geometrically integral generic

^fibre.

Suppose

^that

the

following

conditions hold:

(2) PicXj(

^is

torsion-free,

^Br

XK

^is

^finite,

^and

^XK

^{has a}

(3)

^{either Br}

Xk(,) - ^0,

^or there exists a vertical module M associated to

f

^{such that}

M)

^{= 0.}

If the Manin obstruction is the

only

obstruction to the Hasse

principle

or to weak

approximation

for the k-fibres of

f

^{in a dense} open subset of

P’,

^{then the same}

^property

holds for X.

This theorem should be

compared

with Thm. A of

[CSOO].

^Under

additional

cohomological assumptions

^we

replace

the arithmetic

hypothesis

that the k-fibres of

f satisfy

^{the Hasse}

principle

^{or weak}

approximation by

a weaker condition. The

hypothesis (2)

of the theorem is satisfied when

XK

is a

rationally

^connected

variety.

^{In this case} the existence of a

on

XK

follows from a recent theorem of

Graber,

Harris and Starr

[GHS].

Let us describe the idea of

proof

of Theorem 2.2. Recall that

is the set of adelic

points orthogonal

^{to Br U with}

respect

^to the Brauer- Manin

pairing.

^{As in}

[S96]

^and

[CSOO]

^{we can lift an adelic}

point

of to an adelic

point

^{on some vertical torsor} Y - U. The

difficulty pointed

^{out in}

[H97], 4.5,

^{is that in our} situation there is no reason

why

^{Y should}

satisfy

^{the Hasse}

principle:

^the

hypothesis

^{that the}

Manin obstruction to the Hasse

principle

^{is the}

only

^one for the k-fibres of

f

^{is too} weak for this.

However, using Proposition

2.4 below and the results of Section 1 we show that our condition

(3) implies

^that

IPl

lifts to an adelic

point

^{E where Z is a} relative smooth

compactification

^{of Y. On the other}

hand,

^{we show that on} Z the Manin obstruction to the Hasse

principle

^{is the}

only

^{one: this is a} consequence of the

hypothesis

² and the results of

[H97] (which ^replace

^Thm.

2.1 of

[S96]).

Therefore the

assumption 0 implies

^that

Y(k)

is not

empty,

^hence

U(k)

^is

non-empty.

^{As is dense in}

PROPOSITION 2.4. ^- Let X be a smooth and

geometrically integral variety

^{over a} number field

k,

^{such that}

Gm ) = l~*

^{and Pic X is an}

abelian _group of finite

type.

^{Let T be a}

k-torus,

^{and T be an}

injection

^{of r -}

modules

T

^--+ Pic X.

Suppose that ~ Mv ~

^{E .} Then there exists

a torsor

f :

^{Y - X under T of}

type

T, and a

point

^E

such that

Mv

⁼

f (Pv )

^{for all}

places

^v.

Proof. Let S be the

k-group

^of

multiplicative type

^{dual to the h-} module

Pic X,

5’ 2013~ T be the

surjective k-homomorphism

^dual to T, and

S’1

^{be its} kernel. We denote

by

Id the canonical

isomorphism S -

Pic X.

Under our

assumptions

^{the main theorem} of the descent

theory ([CS87a], [SOl], 6.1.2)

^{states that an adelic}

point

^{on X}

orthogonal

^to

Br, X

^with

respect

^to the Brauer-Manin

pairing

^{can be lifted to an adelic}

point

on some universal torsor Z ---&#x3E; X of

type

Id. Then the

quotient

^{Y = is}

an X-torsor under T of

type

^T

(this trivially

follows from the

functoriality

of

type

^with

respect

^{to the structure}

group).

Let us prove that Z - Y is a universal torsor over Y. Since Z is a universal X-torsor and

= 1~*,

^{we see from}

(4)

^that

HO(Z,Gm)

⁼

^k*

^and

^{Pic Z}

^{= 0. The same} exact sequence

(4)

^considered

for the torsor Z - Y now shows that T - Pic Y is an

isomorphism,

^hence

Z is a universal Y-torsor.

Now let be the

image

^{of on Y. The inverse main theorem}

of the descent

theory ([S01], 6.1.2)

^{states that the}

^image

^{of an adelic}

^point

on a universal torsor is

orthogonal

^to

Bri

^{Y. This is}

exactly

^{what we wanted}

to prove. D

Note that this statement is non-trivial in the

following

^{sense: in}

general

the natural _map

Bri

^{X -}

Bri

^{Y is not}

surjective.

We shall need the

following slight

refinement of

[H97],

Thm. 3.2.1.

PROPOSITION 2.5. - Let F be a closed subset

of A’

of codimension at least

2,

^{L =}

k(A’).

^{Let V be a}

quasi-projective k-variety equipped

^with

a

surjective morphism

p : V ~

A~ B

^{F with}

^split

fibres and

geometrically integral generic

^fibre

VL.

^Let

XL

^{be a} smooth and

projective

^model

of VL.

Assume the

following

conditions:

(a)

The restriction

of p

^{to the}

generic point

^{of a}

sufhciently general

line D in

A’

^{has a smooth}

k(D)-point.

(b)

^Br

XL is finite,

^{and Pic}

(XL) torsion-free,

^{where L is an}

algebraic

closure

of L,

^and

XL = ^XL

^{X L L.}

(c)

There exists a

non-empty

open subset Q

of An

such that for each 9 e

SZ(1~),

the Manin obstruction to the Hasse

principle (resp.

^{to weak}

approximation)

^{is the}

^only

^one for smooth and

projective

models of the fibre

Ve .

Then we have the

following

statements:

(i)

The Manin obstruction to the Hasse

principle (resp.

^to

weak

approximation)

^{is the}

^only

^{one for a} smooth and

projective

^{model V of}

V.

(ii)

^Assume further that

0.

^{Let Q’} be _any

non-empty

open subset

of An.

Then there exists a

k-point 9

^{of 0’ such that a smooth and}

projective

model of the fibre

Ve

^{has a}

k-point.

In

particular, if VL

is smooth and

projective,

^and

0,

^then

p(V(k))

^is Zariski dense in

An

Proof -

(i)

^is

^precisely [H97],

Thm. 3.2.1. We _prove

(ii) using

^the

same induction

argument.

^{As in the}

proof

^of

Prop

^{3.1.1 of}

[H97]

^{we embed}

V into a

projective k-variety Vi,

and consider the closure V’ of the

graph of p in VI xk (A~ B F).

^{We obtain a}

projective morphism p’ :

^{V’ -~}

A’ B F

whose

generic

^fibre

VL

^is

birationally equivalent

^to

VL.

^In

particular, p’

satisfies the

assumptions (a), (b)

^and

(c)

^{of the}

proposition.

^{The fibres of}

p’

^are

split

^{because a}

k-variety containing

^a

split non-empty

open subset is

split.

^{In the}

_{case_n -}

^{1 it follows}

immediately

^from

[H97], proof

^of

Prop. 3.1.1,

^{that if}

0,

^then any

non-empty

open subset of

A)

contains a

k-point

such that the

corresponding

fibre has smooth

k-points.

This _proves

(ii)

^{for n = 1}

^by

Nishimura’s lemma.

Assume that

(ii)

^is

proved

^for any

integer

^{less than n. As in}

[H97], proof

^{of Thm.}

3.2.1,

^{we can find a}

k-point Mo

ⁱⁿ

An

such that the k-

morphism

^obtained

by composing p’

^{with the}

variety satisfying

^the

following properties:

the

generic

fibre of the restriction

pD :

^{is smooth}

and has a

k(D)-point;

o the

geometric generic

^fibre

of pD

^has finite Brauer _group and torsion- free Picard _group.

Since all the fibres

of pD

^are

^split (because

^{the same}

^property

^{holds for}

p’),

^{we can}

apply

^{the case n = 1 of}

Proposition

^2.5

(ii)

^to

p’n. (Recall

^that

0.)

We obtain that each

non-empty

open subset of D contains

a k-rational

point 0

^{such that a} smooth and

projective

^model of the fibre

p’-1 (B)

^{has a}

k-point.

^This proves

(ii)

^for

p’,

^{hence for} p

by

Nishimura’s

lemma. 0

Remark 2.6. ^-

Although

^{we are}

only

interested in the conclusion of

Proposition

2.5 when the

generic

^{fibre is} smooth and

projective,

^{we need}

a more

general

^statement for the induction to work.

Proof of Theorem 2.2. ^- The case 1 is known

([H97], Prop.

3.1.1,

here condition

(3)

^{is not}

required).

^We now assume that

rk f

^{= 2.}

If B ⁼

0,

^{then we take} any U

satisfying

^the conditions of Lemma 2.1. In this case Br U is trivial since it is a

subgroup

^{of Br}

by

Grothendieck’s theorem.

If 0 we choose a vertical module M associated to

f

^with

M)

⁼

0,

^{which is}

possible by

^condition

(3).

^{In this case we define U}

to be the dense _open subset of X such that M ⁼

Mu.

We observe that

k [U] *

^{= k*.}

^Indeed,

^the

generic

^{fibre of}

U -~

PI

^{is proper and}

geometrically integral,

^hence every

regular

^function

on comes from a rational function on

P~. ^However,

^{if such a function}

is

regular

^on

U,

^{then it must be constant due to the}

surjectivity

^{of the}

morphism

^{U -}

Pl.

k

*